Physical realizability

For this equation to describe a real physical system the order of the right-hand side, M, cannot be greater than the order of the left-hand side, N. This criterion for physical realizability is 

This requirement can be proved intuitively from the following reasoning. Take a case where N = 0 and M = 1. 

This equation says that we have a process whose output x depends on the value of the input and the value of the derivative of the input. Therefore the process must be able to diflerentiate, perfectly, the input signal. But it is impossible for any real system to differentiate perfectly. This would require that a step change in the input produce an infinite spike in the output. This is physically impossible. 

This example can be generalized to any case where Af Af to show that differentiation would be required. Therefore N must always be greater than or equal to M. Laplace-transforming Eq. (9.96) gives 

This is a first-order lead. It is physically unrealizable i.e., a real device cannot be built that has exactly this transfer function. 

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In principle, ideal decouphng eliminates control loop interactions and allows the closed-loop system to behave as a set of independent control loops. But in practice, this ideal behavior is not attained for a variety of reasons, including imperfect process models and the presence of saturation constraints on controller outputs and manipulated variables. Furthermore, the ideal decoupler design equations in (8-52) and (8-53) may not be physically realizable andthus would have to be approximated. 

It should be noted that the transformation of A into B by the A variable may or may not correspond to a physically realizable transformation. The change in free energy between two neighbouring points is then given analogously to (16.21), and the whole change is a sum over such terms. 

Physical realizability is often the most difficult of the above three corollary questions to answer. In general, to answer this question it is necessary to know 1) whether the materials and components required by the engineering design are available and 2) whether the manufacturing (and/or fabrication) techniques and skilled craftsmen needed to fabricate the product are also available. These two assessments are difficult to make because they often involve the projection of future technological developments. Technological developments usually do not occur according to schedule. 

We shall adopt Eqs. (9-510) and (9-511) as the covariant wave equation for the covariant four-vector amplitude 9ttf(a ) describing a photon. The physically realizable amplitudes correspond to positive frequency solutions of Eq. (9-510), which in addition satisfy the subsidiary condition (9-511). In other words the admissible wave functions satisfy... 

Note that in the Lorentz gauge we have to adopt the Gupta-Bleuler quantization scheme, with its indefinite metric in a vector space that contains, in addition to the physically realizable states, unphysical... 

Here we consider aggregation in a physically realizable chaotic flow, the journal bearing flow or the vortex mixing flow described earlier. The computations mimic fast coagulation particles seeded in the flow are convected passively and aggregate upon contact. In this example the clusters retain a spherical structure and the capture radius is independent of the cluster size. 

Like direct synthesis, xc is the closed-loop time constant and our only tuning parameter. The first order function raised to an integer power of r is used to ensure that the controller is physically realizable. 2 Again, we would violate this intention in our simple example just so that we can obtain results that resemble an ideal PID controller. 

Before we blindly try to program Eq. (10-7) into a computer, we need to recognize certain pitfalls. If we write out the transfer functions in (10-7), we should find that GFF is not physically realizable—the polynomial in the numerator has a higher order than the one in the denominator.6... 

It is desirable to "decouple" the system so that each manipulated variable appears to affect only one of the output variables. This requires a matrix upstream of the process which, in response to a change in only one of its input signals, will manipulate all the actual process inputs simultaneously so that only the desired output signal changes. If decoupling is to be accomplished only at steady state conditions this matrix is a set of constants. However, if decoupling is required during transient operation the matrix must contain dynamic transfer functions, some of which may not be physically realizable. 

The closest physical realization of the system examined above would be a polymer such as poly(vinyl fluoride) doped into poly(ethylene) (observing the r resonance) or poly(trifluoroethylene) doped into poly(tetra-fluorethylene), observing H. Neither of these systems is likely to be sufficiently miscible, and there is the possibility of significant proportions of head-to-head linkages, which would dominate the experimental M2 because of the proximity of the spins on adjacent backbone atoms. 

Corporate Research, Underwriters Laboratories, Northbrook, IL Motorola Physical Realization Research Center, Schaumburg, IL... 

AB03, 5is seen as a solution of these two end members. Often only one of the limiting compounds is physically realizable. 

Engineering, Installation, Commissioning, and Validation (EICV) This phase involves the physical realization of the design basis, which is developed in response to process risk identified in an H RA study. The bulk of the work in this phase is not a process... 

The right-hand side of this equation contains functions of lime but no derivatives. This ODE can be integrated by evaluating the right-hand side (the derivative) at each point in time and integrating to get z at the new point in time. Then the new value of X is calculated from the known value of m x = (z b, m)/ag. Differentiation is not required and this transfer function is physically realizable ... 

The lead-lag unit is called a derivative unit, and its step response is sketched in Fig. 9.10. For a unit step change in the input, the output jumps to /a and then decays at a rate that depends on z. So the derivative unit approximates an ideal derivative. It is physically realizable since the order of its numerator polynomial is the same as the order of its denominator polynomial. 

Derive the transfer function between CO(jj and Is this transfer function physically realizable ... 

It is useful to consider the ideal situation. If we could design an ideal controller without any regard for physical realizability, what would the ideal elosed-loop regular and servo transfer functions be Clearly, we would wish a load disturbance to have no effect on the controlled variable. So the ideal closedloop regulator transfer function is zero. For setpoint changes, we would like the controlled variable to track the setpoint perfectly at all times. So the ideal servo transfer function is unity. 

This controller is not physically realizable. The negative deadtime implies that we can change the output of the device D minutes before (he input changes, which is impossible. 

This last case illustrates that the desired closedloop relationship cannot be chosen arbitrarily. You cannot moke a jumbo jet behave like a jet fighter We must select the desired response such that the controller b physically realizable. In this case all we need to do is modiiy the specified closedloop servo transfer function to include the dcadtime. 

The basic idea of IMC is to use a model of the process openloop transfer function in such a way that the selection of the specified closedloop response yields a physically realizable feedback controller. 

However, there are two practical problems with this ideal choice of the feedback controller C, y. First, it assumes that the model is perfect More importantly it assumes that the inverse of the plant model Cmo) physically realizable. This is almost never true since most plants have deadtime and/or numerator polynomials that are of lower order than denominator polynomials. 

So the closcdloop servo transfer function S(, must be chosen such that is physically realizable. 

Now the logical choice of that will make physically realizable is the same as that chosen in Eq. (11.63). 

Clearly the best way to select the dosedloop servo transfer function 5(,) to make C, physically realizable is... 

Find the feedforward-controller transfer functions that will keep Xj> constant, by manipulating R, despite changes in z and F. For what values of parameters are these feedforward controllers physically realizable ... 

The basic idea in multivariable IMC is the same as in single-loop IMC. The ideal controller would be the inverse of the plant transfer function matrix. This would give perfect control. However, the inverse of the plant transfer function matrix is not physically realizable because of deadtimes, higher-order denominators than numerators, and RHP zeros (which would give an openloop unstable controller). 

The impulse sampler is, of course, a mathematical fiction an impulse sampler is not physically realizable. But the behavior of a real sampler and hold circuit is practically identical to that of the idealized impulse sampler and hold circuit. The impulse sampler is used in the analysis of sampled-data systems and in the design of sampled-data controllers because it greatly simplifies these calculations. 

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